conversations listlengths 2 2 |
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[
{
"content": "Among 17 parts, only one is slightly lighter and is considered defective. Using a balance scale, determine the minimum number of times you need to weigh to ensure finding the defective part. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"co... |
[
{
"content": "The angle that has the same terminal side as -30° is what? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "The angle that has the same terminal side as -30° can be found by adding or subtracting multiples of 360° (a full rotation)... |
[
{
"content": "Three strictly increasing sequences\n\\[a_1, a_2, a_3, \\ldots,\\qquad b_1, b_2, b_3, \\ldots,\\qquad c_1, c_2, c_3, \\ldots\\]\nof positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer $n$ , the following conditions hold:\n(... |
[
{
"content": "If a worker's wages are decreased by $30\\%$, what percentage increase is necessary for them to regain their original wage?\nA) $30\\%$\nB) $35\\%$\nC) $40\\%$\nD) $42.86\\%$\nE) $45\\%$ Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"conten... |
[
{
"content": "Points \\( M \\) and \\( N \\) are taken on the diagonals \\( AB_1 \\) and \\( BC_1 \\) of the faces of the parallelepiped \\( ABCD A_1 B_1 C_1 D_1 \\), and the segments \\( MN \\) and \\( A_1 C \\) are parallel. Find the ratio of these segments. Let's think step by step and output the final answe... |
[
{
"content": "In right triangle $ABC$, we have $\\angle A = 30^\\circ$, $\\angle B = 90^\\circ$, and $BC = 12$. Find $AB$ to the nearest tenth. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Firstly, utilizing the fact that $ABC$ is a right tr... |
[
{
"content": "Solve for $x$: $3x + 4x = 600 - (5x + 6x)$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Solve for $x$: $3x + 4x = 600 - (5x + 6x)$. The steps are as follows:\n1. Combine like terms on both sides of the equation: $3x + 4x = 7x$... |
[
{
"content": "In $\\triangle ABC$, the sides opposite to angles A, B, C are respectively a, b, c, and $\\cos C= \\frac {3}{5}$.\n(1) Find the value of $\\sin(C+ \\frac {\\pi}{3})$;\n(2) If $\\overrightarrow {CA} \\cdot \\overrightarrow {CB} = 9$, $a+b=8$, find the value of side c and the area of $\\triangle ABC... |
[
{
"content": "The base of isosceles $\\triangle DEF$ is $40$ and its area is $150$. What is the length of one of the congruent sides? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Given that the base \\( EF \\) of isosceles triangle \\( \\tri... |
[
{
"content": "What's the coefficient of the \\(x^3y^5\\) term in the expansion of \\((x+y)^8\\)? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "```python\nimport math\n\ndef binomial_coefficient(n, k):\n return math.comb(n, k)\n\n# We are l... |
[
{
"content": "$\\sin 20^{\\circ}\\cos 10^{\\circ}-\\cos 160^{\\circ}\\sin 170^{\\circ}=$ Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Given:\n\\[\n\\sin 20^{\\circ} \\cos 10^{\\circ} - \\cos 160^{\\circ} \\sin 170^{\\circ}\n\\]\n\nSince \\( ... |
[
{
"content": "Given that $G$ is the centroid of $\\triangle ABC$ and $\\sin A \\overrightarrow{GA} + \\sin B \\overrightarrow{GB} + \\sin C \\overrightarrow{GC} = \\overrightarrow{0}$, determine the value of $\\angle B$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
... |
[
{
"content": "Given a function $y=f(x)$ ($x \\in \\mathbb{R}$), the slope $k$ of the tangent line at any point $P(x, f(x))$ is $k=(x-2)(x-5)^2$. Find the interval where the function is strictly decreasing. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"c... |
[
{
"content": "Calculate \n(1) $\\sqrt {27} + \\sqrt {3} - \\sqrt {12}$ \n(2) $\\frac {1}{\\sqrt {24}} + \\left| \\sqrt {6} - 3 \\right| + \\left( \\frac {1}{2} \\right)^{-1} - 2016^0$ \n(3) $(\\sqrt {3} + \\sqrt {2})^2 - (\\sqrt {3} - \\sqrt {2})^2$. Let's think step by step and output the final answer withi... |
[
{
"content": "Determine the interval(s) where the function $$y = \\log_{0.8}(-x^2 + x + 6)$$ is strictly increasing. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Determine the function \\( f(x) = \\log_{0.8} g(x) \\), where \\( f(x) \\) is s... |
[
{
"content": "1. For the function $f(x) = x^3 + ax^2 + 3x - 9$, it is known that $f(x)$ has an extreme value at $x = -3$. Find the value of $a$.\n2. Observe the following equations: $1 = 1$, $1 - 4 = -(1 + 2)$, $1 - 4 + 9 = (1 + 2 + 3)$, $1 - 4 + 9 - 16 = -(1 + 2 + 3 + 4)$, ... Based on this pattern, deduce the... |
[
{
"content": "Alex is at the candy store buying chocolates. He wants to buy at least 150 chocolates. He wants to buy the least amount such that he would have exactly $15$ leftover after dividing the chocolates evenly among $17$ people. How many chocolates should Alex buy? Let's think step by step and output the... |
[
{
"content": "Given sets A = {2, 4, 6, 16, 29} and B = {4, 16, 20, 27, 29, 32}, find the set intersection of A and B. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To find the intersection of sets A and B, we need to identify the elements tha... |
[
{
"content": "Given the function $f(x) = -f'(0)e^x + 2x$, point $P$ lies on the tangent line $l$ to the curve $y=f(x)$ at the point $(0, f(0))$, and point $Q$ lies on the curve $y=e^x$. Find the minimum value of the distance $|PQ|$. Let's think step by step and output the final answer within \\boxed{}",
"ro... |
[
{
"content": "Two congruent spheres are inscribed into opposite trihedral angles of a cube and touch each other. Calculate the volumes of the spheres if the edge length of the cube is $a$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solv... |
[
{
"content": "Let $f(x)=\\sin x+2xf'(\\frac{\\pi}{3})$, where $f'(x)$ is the derivative of $f(x)$. Find $f'(\\frac{\\pi}{2})$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve the problem, we need to differentiate the given function and... |
[
{
"content": "Given a complex number $z=1+ai$ ($a\\in\\mathbb{R}$), and $z(2+3i)$ is an imaginary number, calculate the value of $a$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Given the complex number $z=1+ai$ and the product $z(2+3i)$ is... |
[
{
"content": "A three-digit natural number has digits $a$, $b$, and $c$ in the hundreds, tens, and units place, respectively, where $a > b$ and $b < c$, and $a$, $b$, and $c$ are distinct elements from the set $\\{1,2,3,4\\}$. Determine the number of such concave numbers. Let's think step by step and output the... |
[
{
"content": "The domain of the function $y=\\log(x^2-2x)$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To find the domain of the function \\( y = \\log(x^2 - 2x) \\), we need to determine for which values of \\( x \\)... |
[
{
"content": "Find the value of $d$ such that the line $y = 3x + d$ is tangent to the parabola $y^2 = 12x$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "```python\nimport sympy as sp\r\n\r\n# Define the symbols\r\nx, d = sp.symbols('x d')\r\... |
[
{
"content": "Find the distance between the planes $x - 2y + 2z = 9$ and $2x - 4y + 4z = 18.$ Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Now, evaluating this step-by-step:\n- Distance between the planes $x - 2y + 2z = 9$ and $2x - 4y + 4z ... |
[
{
"content": "There are $n$ sheep and one wolf disguised as a sheep in a field. Some sheep are friends with each other (friendship is mutual). The wolf's goal is to eat all the sheep. Initially, the wolf chooses some sheep to become friends with. Each following day, the wolf eats one of its friends. Each time t... |
[
{
"content": "Given that $||\\vec{a}||=1$, $||\\vec{b}||=\\sqrt{2}$, and $(\\vec{a}-\\vec{b})\\cdot \\vec{a}=0$, find the angle between $\\vec{a}$ and $\\vec{b}$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "The formula for the dot product i... |
[
{
"content": "Calculate the result of $\\frac{2}{{{x^2}-1}}-\\frac{1}{{x-1}}$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "We need to compute the expression $\\frac{2}{x^2-1} - \\frac{1}{x-1}$. To do this, we should first recognize that $x^... |
[
{
"content": "Given three points in space A(0, 2, 3), B(-2, 1, 6), and C(1, -1, 5), find the angle between vectors $\\overrightarrow{AB}$ and $\\overrightarrow{AC}$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Let's break down the problem i... |
[
{
"content": "Suppose $f(z) = a_3z^3 + a_2z^2 + a_1z + a_0$ and $g(z) = b_2z^2 + b_1z + b_0$, where $a_3$ is nonzero, and the degree of $g(z)$ is less than the degree of $f(z)$. Calculate the degree of $f(z) + g(z)$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},... |
[
{
"content": "Calculate the sum of the geometric series $1 + \\left(\\frac{1}{4}\\right) + \\left(\\frac{1}{4}\\right)^2 + \\left(\\frac{1}{4}\\right)^3 + \\dots$. Express your answer as a common fraction. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"c... |
[
{
"content": "Given the functions $f(x)=\\ln x+ \\frac{1}{x}$ and $g(x)=x+ \\frac{1}{x-a}$, if for every $x_1 \\in (0,2]$, there exists an $x_2 \\in (a,+\\infty)$ such that $f(x_1) \\geqslant g(x_2)$, then the range of the real number $a$ is Let's think step by step and output the final answer within \\boxed{}"... |
[
{
"content": "Compute the distance between the parallel lines given by\n\\[\\begin{pmatrix} 3 \\\\ -2 \\end{pmatrix} + t \\begin{pmatrix} 2 \\\\ -5 \\end{pmatrix}\\]\nand\n\\[\\begin{pmatrix} -1 \\\\ 0 \\end{pmatrix} + s \\begin{pmatrix} 2 \\\\ -5 \\end{pmatrix}.\\] Let's think step by step and output the final... |
[
{
"content": "A natural number had the sum of its digits subtracted from it, resulting in 2007. What could the original number have been? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve this problem, we need to understand the relations... |
[
{
"content": "What is the smallest positive integer that leaves a remainder of 2 when divided by 3 and a remainder of 3 when divided by 5? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "```python\n# We need to find the smallest positive intege... |
[
{
"content": "Find the values of $x$, $y$, and $z$ if $3x - y + z = 8$, $x + 3y - z = 2$, and $x - y + 3z = 6$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "We'll use Python to solve the system of linear equations step by step. We'll represe... |
[
{
"content": "In an interview, ten people were given a test consisting of several questions. It is known that any five people together answered all the questions (i.e., for each question, at least one of the five gave a correct answer), but any four did not. What is the minimum number of questions this could ha... |
[
{
"content": "Let $a$ and $b$ be angles such that $\\sin (a + b) = \\frac{1}{2}$ and $\\sin (a - b) = \\frac{1}{4}$. Find $\\frac{\\tan a}{\\tan b}.$ Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve for \\(\\frac{\\tan a}{\\tan b}\\) gi... |
[
{
"content": "Xiaoyang's average score for Chinese and Mathematics in the midterm exam was 96 points, and Mathematics was 8 points higher than Chinese. The score for Chinese was \\_\\_\\_\\_\\_\\_ points, and the score for Mathematics was \\_\\_\\_\\_\\_\\_ points. Let's think step by step and output the final ... |
[
{
"content": "On January 1st, 2000, it was a Saturday. Determine the day of the week on January 1st, 2010. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve this problem, we need to determine the day of the week for January 1st, 2010, gi... |
[
{
"content": "Solve the system of equations\n$$\n\\left\\{\\begin{array}{l}\nxyz = x + y + z, \\\\\nyzt = y + z + t, \\\\\nztx = z + t + x, \\\\\ntxy = t + x + y.\n\\end{array}\\right.\n$$ Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solv... |
[
{
"content": "Given that $(1-x)^{5}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{5}x^{5}$, find the value of $(a_{0}+a_{2}+a_{4})(a_{1}+a_{3}+a_{5})$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve this problem, we can use the properties of ... |
[
{
"content": "Given the ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$) with eccentricity $\\frac{\\sqrt{3}}{2}$ and points A($a$, 0), B(0, $b$), O(0, 0), where the area of triangle $\\triangle OAB$ is 1,\n1. Find the equation of ellipse $C$.\n2. Find the equation of the line with slope 2 tha... |
[
{
"content": "Evaluate \\( 2(x - a + 5) \\) if the value of \\( x \\) is \\( 2a + 6 \\). Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve the problem, follow this process:\n1. Substitute \\( x = 2a + 6 \\) into \\( 2(x - a + 5) \\).\n2.... |
[
{
"content": "For any positive integer \\( n \\), consider the function \n\\[ g(n) =\\left\\{\\begin{matrix} \\log_{4}{n}, & \\text{if } \\log_{4}{n} \\text{ is rational,} \\\\ 0, & \\text{otherwise.} \\end{matrix}\\right. \\]\nWhat is \\( \\sum_{n = 1}^{2023}{g(n)} \\)?\nA) \\( \\frac{55}{2} \\) \nB) \\( \\fr... |
[
{
"content": "Elmo makes \\( N \\) sandwiches for a fundraiser. For each sandwich, he uses \\( B \\) globs of peanut butter at \\( 4 \\) cents each, \\( J \\) blobs of jam at \\( 5 \\) cents each, and \\( 2 \\) slices of bread at \\( 10 \\) cents per slice. The total cost of the ingredients to make all the sand... |
[
{
"content": "Find the total surface area of a prism circumscribed about a sphere, if the area of its base is \\( S \\). Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "'C purse\nSEntr uurn,\n\nLet's reason through the problem step by step.\n\n... |
[
{
"content": "If the normal vectors of two planes \\\\(\\alpha\\\\) and \\\\(\\beta\\\\) are respectively \\\\(n=(1,0,1)\\\\) and \\\\(\\nu=(-1,1,0)\\\\), then the acute dihedral angle formed by these two planes is \\_\\_\\_\\_\\_\\_\\_\\_ degrees. Let's think step by step and output the final answer within \\b... |
[
{
"content": "Simplify and rationalize the denominator: $$\\frac{1}{1+ \\frac{1}{\\sqrt{5}+2}}.$$ Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To rationalize the denominator, we start by simplifying the complex expression step by step. Given... |
[
{
"content": "Given the function $f(x) = |x - 3|$.\n\n$(1)$ If $f(t) + f(2t) < 9$, find the range of $t$.\n\n$(2)$ If there exists $x \\in [2, 4]$ such that $f(2x) + |x + a| \\leqslant 3$ holds, find the range of $a$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
}... |
[
{
"content": "Calculate: \n1. $(1)(5-6i)+(-2-2i)-(3+3i)$ \n2. $(- \\frac {1}{2}+ \\frac { \\sqrt {3}}{2}i)(2-i)(3+i)$ \n3. $\\frac {(\\sqrt {2}+ \\sqrt {2i})2(4+5i)}{(5-4i)(1-i)}$ Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Calculate:\n1... |
[
{
"content": "Given a cube, among its eight vertices, four vertices form a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve this... |
[
{
"content": "Determine the last digit of the product of all even natural numbers that are less than 100 and are not multiples of ten. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Determine the last digit of the product of all even natural n... |
[
{
"content": "Find the greatest common divisor of $48576$ and $34650$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "'Cuz I am just assisting the user\n 48576 = 2^8 * 3^2 * 7 * 11\n 34650 = 2 * 3^2 * 5^2 * 7 * 11\nSo the greatest common divis... |
[
{
"content": "Given vectors $a=(2,k)$ and $b=(1,1)$, satisfying $b \\perp (a-3b)$.\n\n(Ⅰ) Find the value of $k$;\n\n(Ⅱ) Find the cosine of the angle between vector $a$ and vector $b$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve thi... |
[
{
"content": "Given a hyperbola centered at the origin with foci on the x-axis and eccentricity $e=\\frac{5}{3}$, which is tangent to the line $8x + 2\\sqrt{7}y - 16 = 0$. Find the equation of the hyperbola. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
... |
[
{
"content": "To implement the fundamental task of cultivating morality and nurturing people, and adhere to the comprehensive development of quality education with the \"five educations\" in mind, a school advocates that parents guide their children to do some household chores at home. In order to understand th... |
[
{
"content": "\\[ \\cos 8x + 3 \\cos 4x + 3 \\cos 2x = 8 \\cos x \\cos^3 3x - 0.5 \\] Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "```python\nimport sympy as sp\n\n# Define the variable\nx = sp.symbols('x')\n\n# Define the left-hand side (LH... |
[
{
"content": "The second and fifth terms of a geometric sequence are 18 and 1458, respectively. What is the first term of the sequence? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To find the first term of the geometric sequence, we can fol... |
[
{
"content": "In the right triangular prism $ABC-A_{1}B_{1}C_{1}$, $\\angle BAC=90^{\\circ}$, $BC=2$, $CC_{1}=1$, the angle between line $BC_{1}$ and plane $A_{1}ABB_{1}$ is $60^{\\circ}$. Find the lateral area of the prism $ABC-A_{1}B_{1}C_{1}$. Let's think step by step and output the final answer within \\box... |
[
{
"content": "Jessica completed six assignments with scores of 85, 90, 93, 87, 92, and 95. To calculate her final score, her instructor decides to drop the lowest score and then calculate the arithmetic mean of the remaining scores. What is Jessica's adjusted mean score? Let's think step by step and output the ... |
[
{
"content": "In the diagram, $AB$ and $CD$ are straight lines. What is the value of $x?$ [asy]\ndraw((0,0)--(14,0));\ndraw((0,5)--(14,5));\ndraw((4,0)--(6,5)--(11,0));\nlabel(\"$45^\\circ$\",(6,4.5),W);\nlabel(\"$60^\\circ$\",(6.5,4.5),E);\nlabel(\"$A$\",(0,5),W);\nlabel(\"$C$\",(0,0),W);\nlabel(\"$B$\",(14,5)... |
[
{
"content": "Let $A=\\{1,2,3,\\ldots,40\\}$ . Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\\ne b+c$ . Let's think step by step and output the final a... |
[
{
"content": "If \\( x = 5 \\), what is the value of \\( 3x + 4 \\)? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Now that evaluating the simple expression \\( 3x + 4 \\) with \\( x = 5 \\) has been broken down.\nWe simply substitute \\( x =... |
[
{
"content": "Triangle \\(ABC\\) is isosceles with \\(AB = AC\\) and \\(BC = 65 \\, \\text{cm}\\). \\(P\\) is a point on \\(BC\\) such that the perpendicular distances from \\(P\\) to \\(AB\\) and \\(AC\\) are \\(24 \\, \\text{cm}\\) and \\(36 \\, \\text{cm}\\), respectively. Find the area of \\(\\triangle ABC\... |
[
{
"content": "Given the polynomial $f(x) = x^{2009} + 19x^{2008} + 1$ with distinct zeroes $r_1, \\ldots, r_{2009}$, a polynomial $Q$ of degree $2009$ satisfies \n\\[Q\\left(r_j + \\frac{1}{r_j}\\right) = 0\\] for $j=1, \\ldots, 2009$. Determine the value of $\\frac{Q(1)}{Q(-1)}$. Let's think step by step and o... |
[
{
"content": "The sequence $\\left\\{x_{n}\\right\\}$ is defined as follows: $x_{1}=\\frac{1}{2}, x_{n+1}=x_{n}^{2}+x_{n}$. Find the integer part of the following sum: $\\frac{1}{1+x_{1}}+\\frac{1}{1+x_{2}}+\\cdots+\\frac{1}{1+x_{200}}$. Let's think step by step and output the final answer within \\boxed{}",
... |
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